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Mixed Killing vector field and almost coKähler manifolds

Paritosh Ghosh

TL;DR

This paper introduces mixed Killing vector fields on $(semi)$-Riemannian manifolds, defined by $L_VL_Vg=fL_Vg$, and derives a central curvature identity that yields a Bochner-type rigidity result. It then specializes to almost coKähler geometry, showing that the Reeb field $\xi$ is mixed Killing if and only if the $h$-tensor vanishes, which forces a local product structure, and in dimension three leads to a complete classification. The authors further characterize $\eta$-Einstein and $(\kappa,\mu)$-almost coKähler manifolds under the mixed Killing condition, providing explicit models to illustrate both possible and obstructed cases. These results connect symmetry-driven dynamics with curvature and contact-geometric structures, enabling concrete low-dimensional classifications and guiding future investigations.

Abstract

A vector field $V$ on any (semi-)Riemannian manifold is said to be mixed Killing if for some nonzero smooth function $f$, it satisfies $L_VL_Vg=fL_Vg$, where $L_V$ is the Lie derivative along $V$. This class of vector fields, as a generalization of Killing vector fields, not only identify the isometries of the manifolds, but broadly also contain the class of homothety transformations. We prove an essential curvature identity along those fields on any (semi-)Riemannian manifold and thus generalize the Bochner's theorem for Killing vector fields in this setting. Later we study it in the framework of almost coKähler structure and we prove that the Reeb vector field $ξ$ on an almost coKähler manifold is mixed Killing if and only if the operator $h=0$. Moving further, we completely classify almost coKähler manifolds with $ξ$ mixed Killing vector field in dimension 3. In particular, if $ξ$ on an $η$-Einstein almost coKähler manifold is mixed Killing, then the manifold is of constant scalar curvature with $h=0$. Also we show that on any $(κ,μ)$-almost coKähler manifold, $ξ$ is mixed Killing if and only if the manifold is coKähler. In the end we present few model examples in this context.

Mixed Killing vector field and almost coKähler manifolds

TL;DR

This paper introduces mixed Killing vector fields on -Riemannian manifolds, defined by , and derives a central curvature identity that yields a Bochner-type rigidity result. It then specializes to almost coKähler geometry, showing that the Reeb field is mixed Killing if and only if the -tensor vanishes, which forces a local product structure, and in dimension three leads to a complete classification. The authors further characterize -Einstein and -almost coKähler manifolds under the mixed Killing condition, providing explicit models to illustrate both possible and obstructed cases. These results connect symmetry-driven dynamics with curvature and contact-geometric structures, enabling concrete low-dimensional classifications and guiding future investigations.

Abstract

A vector field on any (semi-)Riemannian manifold is said to be mixed Killing if for some nonzero smooth function , it satisfies , where is the Lie derivative along . This class of vector fields, as a generalization of Killing vector fields, not only identify the isometries of the manifolds, but broadly also contain the class of homothety transformations. We prove an essential curvature identity along those fields on any (semi-)Riemannian manifold and thus generalize the Bochner's theorem for Killing vector fields in this setting. Later we study it in the framework of almost coKähler structure and we prove that the Reeb vector field on an almost coKähler manifold is mixed Killing if and only if the operator . Moving further, we completely classify almost coKähler manifolds with mixed Killing vector field in dimension 3. In particular, if on an -Einstein almost coKähler manifold is mixed Killing, then the manifold is of constant scalar curvature with . Also we show that on any -almost coKähler manifold, is mixed Killing if and only if the manifold is coKähler. In the end we present few model examples in this context.

Paper Structure

This paper contains 9 sections, 13 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. On mixed Killing vector field
  3. Almost coKähler manifolds: An overview
  4. Almost coKähler manifolds with $\xi$ mixed Killing
  5. $\eta$-Einstein almost coKähler manifold with $\xi$ mixed Killing
  6. $(\kappa,\mu)$-almost coKähler manifold with $\xi$ mixed Killing
  7. Collective examples
  8. Conclusions
  9. Acknowledgments

Key Result

Theorem 1

Let $V$ be a Killing vector field on a compact Riemannian manifold $M$ and along $V$ Ricci curvature is $\operatorname{Ric}(V,V)\le0$, then $V$ is parallel. In fact, if Ricci curvature is negative definite, then Killing vector fields are nothing but zero vector field.

Theorems & Definitions (29)

  • Theorem 1
  • Definition 1
  • Theorem 2
  • proof
  • Proposition 3
  • Remark 1
  • Theorem 4
  • proof
  • Remark 2
  • Theorem 5
  • ...and 19 more